Problem 29 - Prealgebra

Question

Find the LCD for each of the following; then use the methods developed in this section to add or subtract as indicated. $$\frac{x}{4}+\frac{1}{5}$$

Step-by-Step Solution

Answer

The sum is \( \frac{5x + 4}{20} \).

Step 1: Identify the Denominators

The given problem is to add \( \frac{x}{4} \) and \( \frac{1}{5} \). The denominators of these fractions are 4 and 5, respectively.

Step 2: Find the Least Common Denominator (LCD)

To find the Least Common Denominator (LCD), we look for the smallest multiple that is common to both denominators (4 and 5). The prime factorizations are: \( 4 = 2^2 \) and \( 5 = 5^1 \). The LCD is the product of the highest powers of all prime factors involved: \( 2^2 \times 5^1 = 20 \). So, the LCD is 20.

Step 3: Rewrite Fractions with the LCD

Convert each fraction to an equivalent fraction with a denominator of 20. For \( \frac{x}{4} \), multiply the numerator and denominator by 5 to get \( \frac{5x}{20} \). For \( \frac{1}{5} \), multiply the numerator and denominator by 4 to get \( \frac{4}{20} \).

Step 4: Add the Fractions

Now that the fractions have a common denominator, we can add them: \( \frac{5x}{20} + \frac{4}{20} = \frac{5x + 4}{20} \).

Step 5: Verify and Simplify if Possible

Confirm that both fractions were correctly rewritten and combined using the addition operation. The final result, \( \frac{5x + 4}{20} \), is already simplified because \( 5x + 4 \) and 20 have no common factors other than 1.

Key Concepts

Prime Factorization

Prime factorization involves breaking down a number into its basic building blocks - the prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. This is a useful strategy when determining the Least Common Denominator (LCD) in fraction addition or subtraction.
For example, to find the LCD of the numbers 4 and 5 as seen in our problem, we need to perform prime factorization:

• 4 is expressed as \(2^2\).
• 5 is already a prime number, expressed as \(5^1\).

To determine the LCD, we take the highest power of each prime present in the factorization of both denominators. Therefore, the LCD is \(2^2 \times 5^1 = 20\).
This approach ensures that the fractions share a common denominator, making addition or subtraction feasible.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. For example, \(\frac{1}{2}\) and \(\frac{2}{4}\) are equivalent because both represent the same part of a whole. This skill is crucial when adjusting fractions to have a common denominator, which is essential for addition or subtraction.
In the exercise, we convert \(\frac{x}{4}\) and \(\frac{1}{5}\) using the LCD 20:

• Multiply the numerator and denominator of \(\frac{x}{4}\) by 5 to get \(\frac{5x}{20}\).
• Multiply the numerator and denominator of \(\frac{1}{5}\) by 4 to get \(\frac{4}{20}\).

These steps adjust each fraction to be equivalent fractions with the same denominator (20). This allows us to easily combine them in the next step.

Adding Fractions

To add two fractions together, they must have the same denominator. When fractions have this, you can simply add the numerators while keeping the same denominator. This process is facilitated by finding a common denominator, like in our given problem.
With both fractions now having a denominator of 20:

• \(\frac{5x}{20} + \frac{4}{20}\)

You add the numerators together, combining the fractions into a single fraction:

• \(\frac{5x + 4}{20}\)

This sum represents the combined value of the original expressions. Ensuring the fractions have a common denominator is vital for this smooth addition.

Simplifying Fractions

After performing operations like addition or subtraction, it's important to check if the resulting fraction can be simplified. Simplifying involves reducing a fraction to its smallest possible form while maintaining its original value.
To simplify a fraction, check if the numerator and the denominator have any common factors other than 1. In our example, the resultant fraction is \(\frac{5x + 4}{20}\). Since 5x + 4 does not have any common factors with 20, the fraction is already in its simplest form.
Simplifying fractions not only makes them easier to read and understand, but also ensures that they form a reduced expression representing the same ratio or value.